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    Coursera
    1. Discrete Math and Analyzing Social Graphs
    2. Week 5
    3. Degrees
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        Video: LectureGraph Invariants and Graph Isomorphism
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        Quiz: Graph Isomorphism
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    1.
    Question 1

    Seven schoolchildren were playing a chess tournament in one round (each plays one game with each). Before the lunch break Ivan played 6 games, Anatoly played 5 games, Alex and Dmitry played 3 games each, Simon and Ilya played 2 games each, and Eugene played only one game. Whom did Alex play with before the lunch break? (Select all correct answers.)

    2 / 2 points
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    Correct
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    Correct
    2.
    Question 2

    For which set of vertex degrees (these are complete lists, no extra vertices allowed) a graph with such degrees does not exist? (Select all correct answers.)

    5 / 5 points
    Check
    Correct

    Such a graph would violate Handshaking Lemma: the sum of degrees here is 121, which is odd.

    Check
    Correct

    Though Handshaking Lemma is satisfied (1+2+2+3+5+5=18 is even), there is no graph with such vertex degrees. (We do not consider multigraphs or pseudographs here.) The reason is as follows: the total number of vertices is 6, thus the two vertices of degree 5 should be connected to all other vertices. Contradiction: the vertex of degree 1 could not be connected to two vertices.

    3.
    Question 3

    How many games did the schoolchildren from Question 1 play before lunch? Recall that Ivan played 6 games, Anatoly played 5 games, Alex and Dmitry played 3 games each, Simon and Ilya played 2 games each, and Eugene played only one game.

    3 / 3 points
    11
    Check
    Correct

    The number of edges in a graph (games played) is counted by summing degrees of vertices: 6+5+3+3+2+2+1=22 and dividing by 2.